Cauchy Problem for Stochastic Nonlinear Schrödinger Equation with Nonlinear Energy-Critical Damping

Abstract

We consider the Cauchy problem for the stochastic nonlinear Schrödinger equation augmented by nonlinear energy-critical damping term arising in nonlinear optics and quantum field theory. Through examining the behavior of the momentum and energy functionals, we almost surely prove the existence and uniqueness of global solutions with continuous $H^1\left({\mathbb{R}}^d\right)$ valued paths. The results cover either defocusing nonlinearity in the full energy critical and subcritical range of exponents or focusing nonlinearity in the full subcritical range, as in the deterministic case.

1. Introduction and Main Result

Nonlinear Schrödinger (NLS) equations model the propagation of the envelope dynamics of a wave packet in weakly nonlinear dispersive media; see [1,2]. Considering an addition of nonlinear damping in nonlinear Schrödinger equations [3,4], these equations arise in many areas of physics, such as nonlinear optics and quantum field theory. They are generic models for the propagation of a laser pulse within an optical fiber under the influence of additional multi-photon absorption processes in nonlinear optics [5]. While considering the three-body interaction in collapsing Bose–Einstein condensates within the realm of the Gross–Pitaevskii theory, they model the emittance of particles from the condensate involving a quintic nonlinear damping [6,7]. It is reasonable to consider the random effects perturbing the system. A formal way to do so in physics is to consider a Gaussian space–time white noise. However, space–time white noise cannot be treated theoretically in mathematics, so the noise, white in time and colored in space, is adopted instead; see [8,9,10,11,12,13] and the references therein.

In this paper, we consider the strongly energy-critical damped stochastic nonlinear Schrödinger equation driven by the additive noise in the energy-subcritical and critical range

(1)$idu+{\displaystyle \frac{1}{2}}\Delta udt={\lambda \left|u\right|}^{2\sigma }udt-ia{\left|u\right|}^{\alpha }udt+dW$

with the initial data

(2)${\left.u\right|}_{t=0}=u_0,$

where $d\ge 3, \lambda \in \mathbb{R}, a>0, 0<\sigma \le {\displaystyle \frac{2}{d-2}}, \alpha ={\displaystyle \frac{4}{d-2}}, t\ge 0$ and $x\in {\mathbb{R}}^d$. We consider the complex valued Wiener process W=${\sum }_{i\in N}{\beta }_i(t,\omega )\varphi e_i\left(x\right)$, where ${\left({\beta }_i\right)}_{i\in N}$ is a sequence of real independent Brownian motions on ($\Omega$,$\mathcal{F}$,$\mathbb{P}$,${\left({\mathcal{F}}_t\right)}_{t\ge 0}$), $\varphi$ is assumed to be an element of $L_2^{0,s}\left({\mathbb{R}}^d\right)$ which is the space of the Hilbert–Schmidt operators from $L^2\left({\mathbb{R}}^d\right)$ into $H^s\left({\mathbb{R}}^d\right)$ endowed with the norm ${\left|\varphi \right|}_{L_2^{0,s}}^2=tr\left({\varphi }^{*}\varphi \right)={\sum }_{i\in N}{\left|\varphi e_i\right|}_{H^s}^2$; ${\left(e_i\right)}_{i\in N}$ is any orthonormal basis of $L^2\left({\mathbb{R}}^d\right)$.

In recent years, the well-posedness of stochastic nonlinear Schrödinger equations has received a great deal of attention. Ref. [8] first investigates stochastic classical NLS equations in the energy space $H^1\left({\mathbb{R}}^d\right)$, it shows the global well-posedness for stochastic classical NLS equations, respectively, perturbed by the additive noise and the conservative multiplicative noise, mainly based on the truncated argument and the behavior of the momentum and energy functionals. For the additive case, the authors obtain the global well-posedness for stochastic classical NLS equations when $\lambda =1,0<\sigma <{\displaystyle \frac{2}{{(d-2)}^{+}}}$ or $\lambda =-1,0<\sigma <{\displaystyle \frac{2}{d}}$. Here, ${\displaystyle \frac{2}{{(d-2)}^{+}}}={\displaystyle \frac{2}{d-2}}(resp.\infty )$ with $d\ge 3(resp.d=1,2).$ The authors in [9] use the gauge transformation and the dispersive estimate for a linear case to prove the global well-posedness for the stochastic classical NLS equations with potentials and the conservative multiplicative noise. Ref. [10] is devoted to the stochastic classical NLS eqautions with linear multiplicative Wiener noise, covering both the conservative and the non-conservative cases through the rescaling approach. This approach reduces the stochastic NLS equations to random NLS equations with lower order terms. The authors show the global well-posedness with exponents of the nonlinearity in exactly the same range as in the deterministic case, i.e., $\lambda =1,0<\sigma <{\displaystyle \frac{2}{{(d-2)}^{+}}}$ or $\lambda =-1,0<\sigma <{\displaystyle \frac{2}{d}}$. Ref. [11] considers the stochastic NLS equations with the weak damping in the defocusing mass-(super)critical range. As for the additive noise, ref. [12] proves that the solutions of defocusing ($\lambda =1$) stochastic classical NLS equations are global when $\sigma ={\displaystyle \frac{2}{d}}$(mass-critical), $d\ge 1$ or $\sigma ={\displaystyle \frac{2}{d-2}}$(energy-critical), $3\le d\le 6$. Ref. [13] obtains the global space–time bound for defocusing mass-critical classical NLS equations with a small multiplicative noise.

In this paper, we consider the global well-posedness for stochastic NLS equation perturbed by the additive noise with an energy-critical damping term. To this end, we first give the complete proof for the local existence of the solutions for Equation (1) through the standard fixed point iteration argument, where the additive noise is additionally treated by the Burkholder inequality. Then, by virtue of the Itô formula, we obtain the evolution laws of the momentum and modified energy. Due to the addition of the nonlinear energy-critical dissipative term, there will be an intersect term of two nonlinearities. We overcome this problem mainly via the certain interpolation. Finally, the global well-posedness for Equation (1) is achieved by estimating the uniform bound of ${\left|u\right|}_{H^1\left({\mathbb{R}}^d\right)}$ in terms of the evolution laws. Next we list the main result of this paper in the following theorem:

Theorem 1. 

Let $d\ge 3$, $a>0$, $\alpha ={\displaystyle \frac{4}{d-2}}$, $\varphi \in L_2^{0,1}$, $0<\kappa <{\displaystyle \frac{a(d-2)}{\alpha +2}}$ and $u_0$ is a ${\mathcal{F}}_0$-measurable random variable with values in $H^1\left({\mathbb{R}}^d\right)$. Suppose further that

• (i) 

  either $\lambda \ge 0$ and $0<\sigma \le {\displaystyle \frac{2}{d-2}}$,

• (ii) 

  or $\lambda <0$ and $0<\sigma <{\displaystyle \frac{2}{d-2}}$,

then, for every $u_0$, there exist an almost surely stopping time ${\tau }^{*}\left(u_0\right)$ and a unique global solution u to Equation (1) with continuous $H^1\left({\mathbb{R}}^d\right)$ valued paths, i.e., ${\tau }^{*}\left(u_0\right)=+\infty ,$ almost surely.

From the perspective of stochastic case, after adding an additive noise, the exponents for nonlinearity are in the same range as the deterministic case [4]. On the other hand, comparing with the stochastic classical NLS equations, the range for exponents of nonlinearity is extended under the effect of the energy-critical damping. The global well-posedness results cover either defocusing nonlinearity in the full energy-critical and subcritical range of exponents or focusing in the full subcritical range.

The rest of the paper is organized as follows: In Section 2, we present the local existence and uniqueness for the solutions of Equation (1). In Section 3, we dedicate the evolution laws of the momentum and modified energy. In Section 4, the global well-posedness of Equation (1), i.e., Theorem 1, is achieved. Note that the constant C may be different from line to line.

2. Local Existence and Uniqueness

In this section, we first recall some inequalities and refer to [4,11,13] for details. Based on the mild form of Equation (1), we show the local existence and uniqueness of the solution through the standard fixed point iteration argument.

Lemma 1 

(Strichartz’s estimates). Let ($q_0,r_0$), ($q_1,r_1$) and ($q_2,r_2$) be admissible pairs, that is ${\displaystyle \frac{2}{q_i}}=d({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{r_i}})$ and $2\le r_i\le {\displaystyle \frac{2d}{d-2}}$ with $i=0,1,2$. Assume $T>0$, and $S\left(t\right)=e^{{\displaystyle \frac{1}{2}}it\Delta }$ denotes the linear Schrödinger propagator. Then, it follows that

${\left|S(·)f\right|}_{L^{q_0}(0,T;L^{r_0})}\le C(r,d){\left|f\right|}_{L^2},$

and

${\left|{\int }_{(0,T)\cap \{s\le t\}}S(t-s)F\left(s\right)ds\right|}_{L^{q_1}(0,T;L^{r_1})}\le C(r_1,r_2,d){\left|F\right|}_{L^{q_2^{\prime }}(0,T;L^{r_2^{\prime }})},$

for all $f\in L^2$ and $F\in L^{q_2^{\prime }}(0,T;L^{r_2^{\prime }}).$ Note that $q_2^{\prime },r_2^{\prime }$ are the conjugate of $q_2,r_2$, respectively.

Lemma 2 

(Gagliardo–Nirenberg inequality). Let $1\le p,q,r\le \infty$, $j,m$ be the integer satisfying $0\le j<m$. Assume that ${\displaystyle \frac{1}{p}}={\displaystyle \frac{j}{d}}+a({\displaystyle \frac{1}{r}}-{\displaystyle \frac{m}{d}})+{\displaystyle \frac{1-a}{q}}$, where $a\in [{\displaystyle \frac{j}{m}},1]$ (if $r>1$ and $m-j-{\displaystyle \frac{d}{r}}=0$, $a<1$). If $|D^m{u|}_{L^r\left({\mathbb{R}}^d\right)}^a{\left|u\right|}_{L^q\left({\mathbb{R}}^d\right)}^{1-a}<\infty$, then there exists $C=C(d,m,j,a,q,r)$, such that

$|D^j{u|}_{L^p\left({\mathbb{R}}^d\right)}\le C|D^m{u|}_{L^r\left({\mathbb{R}}^d\right)}^a{\left|u\right|}_{L^q\left({\mathbb{R}}^d\right)}^{1-a},$

where $D^j,D^m$ denote the j-th, m-th Fréchet derivative, respectively.

Lemma 3 

(Burkholder inequality). Let β be the standard Brownian motion. Let $\mu >1,p\ge 2$, and F be an $L^p\left({\mathbb{R}}^d\right)$-valued process adapted to the filtration generated by ${\beta }_t$. Then, we have

${\left|\underset{0\le a\le b\le t}{sup}{\left|{\int }_a^{b}F\left(s\right)d{\beta }_s\right|}_{L_x^p}\right|}_{L_{\omega }^{\mu }}\le C(p,\mu ){\left|{\int }_0^t{\left|F\left(s\right)\right|}_{L_x^p}^{2}ds\right|}_{L_{\omega }^{{\displaystyle \frac{\mu }{2}}}}^{{\displaystyle \frac{1}{2}}}.$

Next we show the local well-posedness for Equation (1) in the following theorem.

Theorem 2. 

Assume that $d\ge 3$, $0<\sigma \le {\displaystyle \frac{2}{d-2}}$, $\alpha ={\displaystyle \frac{4}{d-2}}$, $\varphi \in L_2^{0,1}$, $\lambda ,a\in \mathbb{R}$, and $u_0$ is a ${\mathcal{F}}_0$-measurable random variable with values in $H^1\left({\mathbb{R}}^d\right)$. Then, for every $u_0$, there exists a unique solution u to Equation (1) with continuous $H^1\left({\mathbb{R}}^d\right)$ valued paths, which is defined on a random interval $[0,{\tau }^{*}\left(u_0\right))$, where ${\tau }^{*}\left(u_0\right)$ is a stopping time, such that

${\tau }^{*}\left(u_0\right)=+\infty \mathit{or}\underset{t\to {\tau }^{*}\left(u_0\right)}{lim}|u\left(t\right){|}_{H^1\left({\mathbb{R}}^d\right)}=+\infty .$

Proof. 

In this part, we mainly use the following mild form of Equation (1), that is

(3)$u\left(t\right)=S\left(t\right)u_0-i\lambda {\int }_0^t{S(t-s)\left(\right|u|}^{2\sigma }u)\left(s\right)ds-a{\int }_0^t{S(t-s)\left(\right|u|}^{\alpha }u)\left(s\right)ds-i{\int }_0^{t}S(t-s)dW\left(s\right).$

Next we will prove the local existence and uniqueness for the solutions of Equation (1). Denote the right hand of (3) by $\Psi \left(u\right)\left(t\right)$, and define

$X_T=L^{\infty }(0,T;L^2\left({\mathbb{R}}^d\right))\cap L^q(0,T;L^r\left({\mathbb{R}}^d\right))\cap L^{\gamma }(0,T;L^{\rho }\left({\mathbb{R}}^d\right))$

for some $T>0$, where $r=2\sigma +2$, $q={\displaystyle \frac{4\sigma +4}{d\sigma }}$, $\rho ={\displaystyle \frac{2d^2}{d^2-2d+4}}$, $\gamma =\alpha +2$. In the following, we aim to prove that $\Psi \left(u\right)\left(t\right)$ and $\nabla \Psi \left(u\right)\left(t\right)$ have paths in $X_T$, i.e., for any fixed $\omega \in \Omega$, $\Psi \left(u\right)\left(t\right),\nabla \Psi \left(u\right)\left(t\right)\in X_T$ almost surely.

We first prove $\Psi \left(u\right)\left(t\right)\in X_T$ almost surely, based on Equation (3), we have

$\begin{array}{cc}\hfill {|\Psi \left(u\right)\left(t\right)|}_{X_T}\le & |S\left(t\right)u_0{|}_{X_T}+C{\left|{\int }_0^t{S(t-s)(|u|}^{2\sigma }u)\left(s\right)ds\right|}_{X_T}+C{\left|{\int }_0^t{S(t-s)(|u|}^{\alpha }u)\left(s\right)ds\right|}_{X_T}\hfill \end{array}$

$\begin{array}{c}\hfill +C{\left|\sum _{i\in N}{\int }_0^{t}S(t-s)\varphi e_i\left(x\right)d{\beta }_i\left(s\right)\right|}_{X_T}:=I_1+I_2+I_3+I_4.\end{array}$

By Strichartz’s estimates,

$I_1+I_2+I_3\le C(r,\rho ,d)|u_0{|}_{L^2}+{C(r,\rho ,d)\left|\right|u|}^{2\sigma }{u|}_{L^{q^{\prime }}(0,T;L^{r^{\prime }})}+{C(r,\rho ,d)\left|\right|u|}^{\alpha }{u|}_{L^{{\gamma }^{\prime }}(0,T;L^{{\rho }^{\prime }})}.$

According to the definition of $r,q,\rho ,\gamma$, we introduce the following exponents

${\displaystyle \frac{1}{r^{\prime }}}={\displaystyle \frac{2\sigma }{r}}+{\displaystyle \frac{1}{r}};{\displaystyle \frac{1}{q^{\prime }}}={\displaystyle \frac{2\sigma }{{\theta }_1}}+{\displaystyle \frac{1}{q}};{\displaystyle \frac{1}{{\rho }^{\prime }}}={\displaystyle \frac{\alpha }{{\theta }_2}}+{\displaystyle \frac{1}{\rho }};{\displaystyle \frac{1}{{\gamma }^{\prime }}}={\displaystyle \frac{\alpha }{\gamma }}+{\displaystyle \frac{1}{\gamma }},$

where ${\theta }_1={\displaystyle \frac{2\sigma (2\sigma +2)}{2-(d-2)\sigma }}$, ${\theta }_2={\displaystyle \frac{2d^2}{{(d-2)}^2}}$. Using the Hölder inequality,

$I_2+I_3\le {C\left|u\right|}_{L^{{\theta }_1}(0,T;L^r)}^{2\sigma }{\left|u\right|}_{L^q(0,T;L^r)}+{C\left|u\right|}_{L^{\gamma }(0,T;L^{{\theta }_2})}^{\alpha }{\left|u\right|}_{L^{\gamma }(0,T;L^{\rho })}.$

Through the Hölder inequality again and Sobolev embedding,

(4)$I_2\le C{T}^{{\displaystyle \frac{2\sigma }{{\theta }_1}}}{\left|u\right|}_{L^{\infty }(0,T;L^r)}^{2\sigma }{\left|u\right|}_{L^q(0,T;L^r)}\le C{T}^{{\displaystyle \frac{2\sigma }{{\theta }_1}}}{\left|u\right|}_{L^{\infty }(0,T;H^1)}^{2\sigma }{\left|u\right|}_{L^q(0,T;L^r)}.$

Employing Gagliardo–Nirenberg inequality,

(5)$I_3\le {C(\gamma ,\rho ,d)|\nabla u|}_{L^{\gamma }(0,T;L^{\rho })}^{\alpha }{\left|u\right|}_{L^{\gamma }(0,T;L^{\rho })}.$

More treatments are needed on $I_4$. We denote $z(t,x):={\sum }_{i\in N}{\int }_0^{t}S(t-s)\varphi e_i\left(x\right)d{\beta }_i\left(s\right)$, then we have

$\begin{array}{cc}\hfill {\left|z(t,x)\right|}_{X_T}& ={\left|z(t,x)\right|}_{L^{\infty }(0,T;L^2)}+{\left|z(t,x)\right|}_{L^q(0,T;L^r)}+{\left|z(t,x)\right|}_{L^{\gamma }(0,T;L^{\rho })}\hfill \\ \hfill & \le {\left|z(t,x)\right|}_{L^{\infty }(0,T;L^2)}+{C\left|z(t,x)\right|}_{L^{\infty }(0,T;L^r)}+C{\left|z(t,x)\right|}_{L^{\infty }(0,T;L^{\rho })}.\hfill \end{array}$

Since $r,\rho >2$, we can utilize Burkholder inequality and then Strichartz’s estimates,

(6)$\begin{array}{cc}\hfill I_4& \le C{\left({\int }_0^T\sum _{i\in N}\left({\int }_{{\mathbb{R}}^d}{\left|S(t-s)\varphi e_i\left(x\right)\right|}^{2}dx\right)ds\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & +C{\left({\int }_0^T\sum _{i\in N}{\left({\int }_{{\mathbb{R}}^d}{\left|S(t-s)\varphi e_i\left(x\right)\right|}^{r}dx\right)}^{{\displaystyle \frac{2}{r}}}ds\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & +C{\left({\int }_0^T\sum _{i\in N}{\left({\int }_{{\mathbb{R}}^d}{\left|S(t-s)\varphi e_i\left(x\right)\right|}^{\rho }dx\right)}^{{\displaystyle \frac{2}{\rho }}}ds\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le C{\left({\int }_0^{T}S\left(2(t-s)\right){\left|\varphi \right|}_{L_2^{0,0}}^{2}ds\right)}^{{\displaystyle \frac{1}{2}}}\le C\left(T\right){\left|\varphi \right|}_{L_2^{0,0}}.\hfill \end{array}$

Gathering (4)–(6), we have

${\left|\Psi \left(u\right)\left(t\right)\right|}_{X_T}\le C|u_0{|}_{L^2}+C{T}^{{\displaystyle \frac{2\sigma }{{\theta }_1}}}{\left|u\right|}_{L^{\infty }(0,T;H^1)}^{2\sigma }{\left|u\right|}_{L^q(0,T;L^r)}+{C|\nabla u|}_{L^{\gamma }(0,T;L^{\rho })}^{\alpha }{\left|u\right|}_{L^{\gamma }(0,T;L^{\rho })}+C{\left|\varphi \right|}_{L_2^{0,0}}.$

Since the space derivative and $S(·)$ commute, we obtain

$\begin{array}{cc}\hfill {|\nabla \Psi \left(u\right)\left(t\right)|}_{X_T}\le C{|\nabla u_0|}_{L^2}& +C{T}^{{\displaystyle \frac{2\sigma }{{\theta }_1}}}{\nabla \left(\right|u|}_{L^{\infty }(0,T;H^1)}^{2\sigma }{\left|u\right|}_{L^q(0,T;L^r)})\hfill \\ \hfill & +{C\nabla \left(\right|\nabla u|}_{L^{\gamma }(0,T;L^{\rho })}^{\alpha }{\left|u\right|}_{L^{\gamma }(0,T;L^{\rho })}{)+C|\varphi |}_{L_2^{0,1}}.\hfill \end{array}$

Notice that $u_0$ has values in $H^1\left({\mathbb{R}}^d\right)$, there exists $C_1>0$ such that $|\nabla u_0{|}_{L^2}\le C_1$ almost surely; it follows that there exists $C_2>0$, such that $|S(·)\nabla u_0{|}_{X_T}\le C_2$ almost surely for a small enough T. Thus, we achieve $\Psi$ which maps the set

$\begin{array}{ll}\mathcal{B}=& \{{u;|\nabla u|}_{L^{\gamma }(0,T;L^{\rho })}\le 2C_2{,|\nabla u|}_{L^{\infty }(0,T;L^2))\cap L^q(0,T;L^r)}\le 2C\left(\right|\nabla u_0{|}_{L^2}+{|\varphi |}_{L_2^{0,1}}),\\ & {\left|u\right|}_{X_T}\le 2C\left(\right|u_0{|}_{L^2}+{|\varphi |}_{L_2^{0,0}}\left)\right\}\end{array}$

to itself, almost surely, and is a contraction mapping in the $X_T$ norm, provided $C_2$ and T are chosen properly and small. The fixed point iteration theorem directly implies the existence and uniqueness of the solution to Equation (1) on $[0,T]$ almost surely. Finally, considering that $\omega \in \Omega$, Theorem 2 follows. □

3. The Evolution Laws of the Momentum and Modified Energy

In this section, we give the evolution laws of the momentum

(7)$M\left(u\right):={\int }_{{\mathbb{R}}^d}{\left|u\right|}^{2}dx$

and the modified energy

(8)$H\left(u\right):={\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^d}{|\nabla u|}^{2}dx+{\displaystyle \frac{\lambda }{\sigma +1}}{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{2\sigma +2}dx+\kappa {\int }_{{\mathbb{R}}^d}{\left|u\right|}^{\alpha +2}dx,$

where $0<\kappa <{\displaystyle \frac{a(d-2)}{\alpha +2}}$. We present the behavior of these functionals and show a priori estimates of u.

Proposition 1. 

Let $u_0$, σ and ϕ be as in Theorem 2, for any stopping time τ, such that $\tau <{\tau }^{*}\left(u_0\right)$ almost surely, we have

(9) $\begin{array}{c}\hfill M((u(\tau ))=M\left(u_0\right)+2Im\sum _{i\in N}{\int }_0^{\tau }{\int }_{{\mathbb{R}}^d}\overline{u}\varphi e_i\left(x\right)dxd{\beta }_i\left(s\right)-2a{\int }_0^{\tau }{\int }_{{\mathbb{R}}^d}|u{|}^{\alpha +2}dxds+|\varphi {|}_{L_2^{0,0}}^2\tau ,\end{array}$

where u is the solution of Equation (1) given by Theorem 2 with initial data $u\left(0\right)=u_0.$

Proof. 

To obtain the evolution of momentum $M\left(u\right(\tau \left)\right)$, the truncated argument is applied (see [8] Proposition 3.2). Introduce the cut-off function $\theta$ that $\theta$ is a non-negative $C^{\infty }$ function on ${\mathbb{R}}^{+}$ satisfying $\theta \left(x\right)=1$ for $\left|x\right|\le 1$, $\theta \left(x\right)=0$ for $\left|x\right|>2.$ Then, we define for $m\in \mathbb{N}$, the operators ${\Theta }_m$ by

$\mathcal{F}\left({\Theta }_{m}v\right)\left(\xi \right):=\theta \left({\displaystyle \frac{\left|\xi \right|}{m}}\right)\mathcal{F}\left(v\right)\left(\xi \right),\xi \in {\mathbb{R}}^d,$

where $\mathcal{F}$ is the Fourier transform. Let $m=(m_1,m_2,m_3,m_4)\in {\mathbb{N}}^4$ and $R>0$. When $m\to \infty$, we mean that $m_1\to \infty$ first, then $m_2\to \infty$, $m_3\to \infty$ and finally $m_4\to \infty$. We have the following truncated approximation:

(10)$\begin{array}{cc}\hfill id{u}_m^R+{\displaystyle \frac{1}{2}}{\Theta }_{m_1}\Delta u_m^{R}dt-\lambda \theta \left({\displaystyle \frac{|u_m^R{|}_{H^1}}{R}}\right){\Theta }_{m_2}\left(|u_m^R{|}^{2\sigma }{u}_m^R\right)dt+ia\theta \left({\displaystyle \frac{|u_m^R{|}_{H^1}}{R}}\right){\Theta }_{m_3}& \left(|u_m^R{|}^{\alpha }{u}_m^R\right)dt\hfill \\ \hfill & ={\Theta }_{m_4}dW.\hfill \end{array}$

Since Equation (10) is globally Lipschitz continuous, it is known that there exists a unique solution $u_m^R$ defined for $t\ge 0$, satisfying $u_m^R\left(0\right)=u_0$. Then, when $m\to \infty$, $u_m^R$ almost surely converges in $C([0,\tau ];H^1)$ to the unique $u_R$ of

(11)$\begin{array}{c}\hfill id{u}^R+{\displaystyle \frac{1}{2}}\Delta u^{R}dt-\lambda \theta \left({\displaystyle \frac{|u^R{|}_{H^1}}{R}}\right)\left(|u^R{|}^{2\sigma }{u}^R\right)dt+ia\theta \left({\displaystyle \frac{|u^R{|}_{H^1}}{R}}\right)\left(|u^R{|}^{\alpha }{u}^R\right)dt=dW.\end{array}$

For almost every $\omega \in \Omega$, there always exists R, such that ${\left|u\right|}_{X_{\tau }}\le R$ and $u\left(t\right)=u^R\left(t\right)$ on $[0,\tau ]$; thus, we apply the Itô formula to $M\left(u\right(\tau \left)\right)$.

We respectively denote $D,D^2$ as the first and second Fréchet derivative. It holds that

$DM\left(u\right)\left({\phi }_1\right)=2\mathrm{Re}{\int }_{{\mathbb{R}}^d}\overline{u}{\phi }_{1}dx,D^{2}M\left(u\right)({\phi }_1,{\phi }_2)=2\mathrm{Re}{\int }_{{\mathbb{R}}^d}{\phi }_1\overline{{\phi }_2}dx.$

Applying Itô formula, we have

$\begin{array}{cc}\hfill dM\left(u\right)& =DM\left(u\right)\left(du\right)+{\displaystyle \frac{1}{2}}{D}^{2}M\left(u\right)(du,du)\hfill \\ \hfill & =2\mathrm{Im}{\int }_{{\mathbb{R}}^d}\overline{u}dxdW-2a{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{\alpha +2}dxdt+{\left|\varphi \right|}_{L_2^{0,0}}^{2}dt\hfill \end{array}$

which leads to

$\begin{array}{c}\hfill M(\left(u\left(\tau \right)\right)=M\left(u_0\right)+2\mathrm{Im}\sum _{i\in N}{\int }_0^{\tau }{\int }_{{\mathbb{R}}^d}\overline{u}\varphi e_i\left(x\right)dxd{\beta }_i\left(s\right)-2a{\int }_0^{\tau }{\int }_{{\mathbb{R}}^d}|u{|}^{\alpha +2}dxds+|\varphi {|}_{L_2^{0,0}}^2\tau .\end{array}$

□

Lemma 4. 

Let $u_0$, σ and ϕ be as in Theorem 2. For any stopping time τ, such that $\tau <{\tau }^{*}\left(u_0\right)$ almost surely, we determine that for any $k\in \mathbb{N}$, there exists a constant $C\ge 0$, such that

(12) $\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}{M}^k\left(u\left(t\right)\right)\right)\le C\mathbf{E}\left(M^k\left(u_0\right)\right),$

and

(13) $\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}{\int }_{{\mathbb{R}}^d}{\left|u\left(t\right)\right|}^{\alpha +2}dx\right)\le C\mathbf{E}\left(M\left(u_0\right)\right).$

Proof. 

We first study (12) and utilize the Itô formula to obtain the following for $t\in [0,\tau ],$

$\begin{array}{cc}\hfill M^k\left(u\left(t\right)\right)=& M^k\left(u_0\right)+2k\mathrm{Im}\sum _{i\in N}{\int }_0^t{M}^{k-1}\left(u\left(s\right)\right){\int }_{{\mathbb{R}}^d}\overline{u}\left(s\right)\varphi e_i\left(x\right)dxd{\beta }_i\left(s\right)\hfill \\ \hfill & -2ka{\int }_0^t{M}^{k-1}\left(u\left(s\right)\right){\int }_{{\mathbb{R}}^d}{\left|u\left(s\right)\right|}^{\alpha +2}dxds+k{\left|\varphi \right|}_{L_2^{0,0}}^2{\int }_0^t{M}^{k-1}\left(u\left(s\right)\right)ds\hfill \\ \hfill & +2k(k-1){\int }_0^t{M}^{k-2}\left(u\left(s\right)\right)\sum _{i\in N}{\int }_{{\mathbb{R}}^d}{\left(\mathrm{Im}\left(\overline{u}\left(s\right)\varphi e_i\left(x\right)\right)\right)}^{2}dxds.\hfill \end{array}$

Taking the supremum and employing the martingale inequality yields

$\begin{array}{cc}\hfill \mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}{M}^k\left(u\left(t\right)\right)\right)\le & \mathbf{E}\left(M^k\left(u_0\right)\right)+6k\mathbf{E}\left({\left({\int }_0^{\tau }{M}^{2k-2}\left(u\left(s\right)\right){\left|{\varphi }^{*}u\left(s\right)\right|}_{L^2}^{2}ds\right)}^{{\displaystyle \frac{1}{2}}}\right)\hfill \\ \hfill & +{k\left|\varphi \right|}_{L_2^{0,0}}^2\mathbf{E}\left({\int }_0^{\tau }{M}^{k-1}\left(u\left(s\right)\right)ds\right)\hfill \\ \hfill & +2k(k-1)\mathbf{E}\left({\int }_0^{\tau }{M}^{k-2}\left(u\left(s\right)\right){\left|{\varphi }^{*}u\left(s\right)\right|}_{L^2}^{2}ds\right).\hfill \end{array}$

It is known that

$|{\varphi }^{*}{u\left(s\right)|}_{L^2}^2\le |{\varphi }^{*}{|}_{L_2^{0,0}}^2{\left|u\left(s\right)\right|}_{L^2}^2={\left|\varphi \right|}_{L_2^{0,0}}^{2}M\left(u\left(s\right)\right),$

where ${\varphi }^{*}$ is the adjoint operator of $\varphi$ with $|{\varphi }^{*}{|}_{L_2^{0,0}}^2={|\varphi |}_{L_2^{0,0}}^2$. Thus, we have

$\begin{array}{cc}\hfill \mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}{M}^k\left(u\left(t\right)\right)\right)\le & \mathbf{E}\left(M^k\left(u_0\right)\right)+6k{\left|\varphi \right|}_{L_2^{0,0}}\mathbf{E}\left({\left({\int }_0^{\tau }{M}^{2k-1}\left(u\left(s\right)\right)ds\right)}^{{\displaystyle \frac{1}{2}}}\right)\hfill \\ \hfill & +{k(2k-1)\left|\varphi \right|}_{L_2^{0,0}}^2\mathbf{E}\left({\int }_0^{\tau }{M}^{k-1}\left(u\left(s\right)\right)ds\right).\hfill \end{array}$

By the Hölder inequality, we finally achieve

$\begin{array}{cc}\hfill \mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}{M}^k\left(u\left(t\right)\right)\right)\le & \mathbf{E}\left(M^k\left(u_0\right)\right)+6k{T}_1{\left|\varphi \right|}_{L_2^{0,0}}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}{M}^{k-{\displaystyle \frac{1}{2}}}\left(u\left(t\right)\right)\right)\hfill \\ \hfill & +k(2k-1)T_2{\left|\varphi \right|}_{L_2^{0,0}}^2\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}{M}^{k-1}\left(u\left(t\right)\right)\right)\hfill \\ \hfill \le & \mathbf{E}\left(M^k\left(u_0\right)\right)+C(k,T,\varphi )\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}{M}^{k-{\displaystyle \frac{1}{2}}}\left(u\left(t\right)\right)\right).\hfill \end{array}$

Then, through an induction argument, we prove (12). In particular, take $k=1$, (13) follows. □

Remark 1. 

Note that the result of the above lemma can be generalized to $k\ge 0$ and we give a more straightforward version

(14) $\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}{\left|u\left(t\right)\right|}_{L^2}^{2k}\right)\le C\mathbf{E}\left(|u_0{|}_{L^2}^{2k}\right).$

With a priori estimates in the above lemma, we can conclude the uniform bound on the $L^2$-norm of u.

Next, we consider the evolution law of the modified energy functional.

Proposition 2. 

Let $u_0$, σ and ϕ be as in Theorem 2, for any stopping time τ such that $\tau <{\tau }^{*}\left(u_0\right)$ a.s., we have

(15) $\begin{array}{cc}\hfill H\left(u\right(\tau \left)\right)=& H\left(u_0\right)-(a-{\displaystyle \frac{\alpha (\alpha +2)}{4}}\kappa ){\int }_0^{\tau }{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{\alpha }{|\nabla u|}^{2}dxds-Im{\int }_0^{\tau }{\int }_{{\mathbb{R}}^d}\Delta \overline{u}dxdW\hfill \\ \hfill & -{\displaystyle \frac{\alpha (\alpha +2)}{4}}\kappa {\int }_0^{\tau }{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{\alpha }|Re\left({\displaystyle \frac{\overline{u}}{\left|u\right|}}\nabla u\right)-Im\left({\displaystyle \frac{\overline{u}}{\left|u\right|}}\nabla u\right){|}^{2}dxds\hfill \\ \hfill & -a\alpha {\int }_0^{\tau }{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{\alpha }{|\nabla |u\left|\right|}^{2}dxds-2a\lambda {\int }_0^{\tau }{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{2\sigma +\alpha +2}dxds\hfill \\ \hfill & +2\lambda Im{\int }_0^{\tau }{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{2\sigma }\overline{u}dxdW-a(\alpha +2)\kappa {\int }_0^{\tau }{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{2\alpha +2}dxds\hfill \\ \hfill & +(\alpha +2)\kappa Im{\int }_0^{\tau }{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{\alpha }\overline{u}dxdW+{\displaystyle \frac{1}{2}}\sum _{i\in N}{\int }_0^{\tau }{\int }_{{\mathbb{R}}^d}{|\nabla \varphi e_i\left(x\right)|}^{2}dxds\hfill \\ \hfill & +\lambda \sum _{i\in N}{\int }_0^{\tau }{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{2\sigma }{\left|\varphi e_i\left(x\right)\right|}^{2}dxds\hfill \\ \hfill & +2\sigma \lambda \sum _{i\in N}{\int }_0^{\tau }{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{2\sigma -2}{\left(Im\left(\overline{u}\varphi e_i\left(x\right)\right)\right)}^{2}dxds\hfill \\ \hfill & +{\displaystyle \frac{\alpha +2}{2}}\kappa \sum _{i\in N}{\int }_0^{\tau }{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{\alpha }{\left|\varphi e_i\left(x\right)\right|}^{2}dxds\hfill \\ \hfill & +{\displaystyle \frac{\alpha (\alpha +2)}{2}}\kappa \sum _{i\in N}{\int }_0^{\tau }{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{\alpha -2}{\left(Im\left(\overline{u}\varphi e_i\left(x\right)\right)\right)}^{2}dxds,\hfill \end{array}$

where u is the solution of Equation (1) given by Theorem 2 with initial data $u\left(0\right)=u_0$ and $0<\kappa <{\displaystyle \frac{a(d-2)}{\alpha +2}}$.

Proof. 

The proof is similar to that of Proposition 1; by employing the Itô formula to $H\left(u\right)$, the derivatives of $H\left(u\right)$ along directions ${\phi }_1$ and $({\phi }_1,{\phi }_2)$ are as follows

$DH\left(u\right)\left({\phi }_1\right)=\mathrm{Re}{\int }_{{\mathbb{R}}^d}\nabla \overline{u}\nabla {\phi }_{1}dx+2\lambda \mathrm{Re}{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{2\sigma }\overline{u}{\phi }_{1}dx+(\alpha +2)\kappa \mathrm{Re}{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{\alpha }\overline{u}{\phi }_{1}dx,$

$\begin{array}{cc}\hfill D^{2}H\left(u\right)({\phi }_1,{\phi }_2)=& \mathrm{Re}{\int }_{{\mathbb{R}}^d}\nabla \overline{{\phi }_2}\nabla {\phi }_{1}dx+2\lambda \mathrm{Re}{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{2\sigma }\overline{{\phi }_2}{\phi }_{1}dx\hfill \\ \hfill & +4\sigma \lambda {\int }_{{\mathbb{R}}^d}{\left|u\right|}^{2\sigma -2}\mathrm{Re}\left(\overline{u}{\phi }_1\right)\mathrm{Re}\left(\overline{u}{\phi }_2\right)dx+(\alpha +2)\kappa \mathrm{Re}{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{\alpha }\overline{{\phi }_2}{\phi }_{1}dx\hfill \\ \hfill & +\alpha (\alpha +2)\kappa {\int }_{{\mathbb{R}}^d}{\left|u\right|}^{\alpha -2}\mathrm{Re}\left(\overline{u}{\phi }_1\right)\mathrm{Re}\left(\overline{u}{\phi }_2\right)dx.\hfill \end{array}$

Itô formula leads to

$H\left(u\left(\tau \right)\right)=H\left(u_0\right)+{\int }_0^{\tau }\left(DH\left(u\right)\left(du\right)\right)ds+{\displaystyle \frac{1}{2}}{\int }_0^{\tau }\left(D^{2}H\left(u\right)(du,du)\right)ds$

which implies (15); we conclude the proof. □

4. Global Well-Posedness

In this section, we show the global well-posedness of Equation (1), i.e.,Thorem 1 by means of the boundness of the modified energy evolution law.

Based on the modified energy functional (8), we have the following lemma.

Lemma 5. 

Assume that $0<\sigma \le {\displaystyle \frac{2}{d-2}}$, there exists a constant $C(\sigma ,\lambda )$, such that

• (i) 

  if $\lambda \ge 0$, then

  (16) ${\int }_{{\mathbb{R}}^d}{|\nabla u|}^{2}dx\le 2H\left(u\right)\mathrm{and}{\displaystyle \frac{\lambda }{\sigma +1}}{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{2\sigma +2}dx\le H\left(u\right),$

• (ii) 

  if $\lambda <0$ and $0<\sigma <{\displaystyle \frac{2}{d-2}}$, then

  (17) ${\int }_{{\mathbb{R}}^d}{|\nabla u|}^{2}dx\le 2H\left(u\right)+C(\sigma ,\lambda ){\left({\int }_{{\mathbb{R}}^d}{\left|u\right|}^{2}dx\right)}^{\sigma +1}.$

Proof. 

When $\lambda \ge 0$, the case (i) directly follows from the definition of $H\left(u\right)$.

In regard to case (ii), we focus on the relationship between ${\int }_{{\mathbb{R}}^d}{\left|u\right|}^{2\sigma +2}dx$ and ${\int }_{{\mathbb{R}}^d}{\left|u\right|}^{\alpha +2}$-$dx$, through the interpolation and Young inequality, we have

(18)${\left|u\right|}_{L^{2\sigma +2}}^{2\sigma +2}\le {\left|u\right|}_{L^{\alpha +2}}^{(2\sigma +2)\theta }{\left|u\right|}_{L^2}^{(2\sigma +2)(1-\theta )}\le {\epsilon \left|u\right|}_{L^{\alpha +2}}^{\alpha +2}+C\left(\epsilon \right){\left|u\right|}_{L^2}^{2\sigma +2}$

with $\theta ={\displaystyle \frac{d\sigma }{2\sigma +2}}\in (0,1)$ and when $\epsilon$ is small enough, we achieve

$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{|\nabla u|}_{L^2}^2& \le H\left(u\right)+{\displaystyle \frac{\left|\lambda \right|}{\sigma +1}}\left({\epsilon \left|u\right|}_{L^{\alpha +2}}^{\alpha +2}+C\left(\epsilon \right){\left|u\right|}_{L^2}^{2\sigma +2}\right)-\kappa {\left|u\right|}_{L^{\alpha +2}}^{\alpha +2}\hfill \\ \hfill & \le H\left(u\right)+{C(\sigma ,\lambda )\left|u\right|}_{L^2}^{2\sigma +2}.\hfill \end{array}$

Then, the case (ii) follows. □

With Lemma 5 in hand, we proceed to estimate $\mathbf{E}\left({sup}_{t\in [0,\tau ]}H\left(u\left(t\right)\right)\right)$.

Lemma 6. 

Let $u_0$, σ and ϕ be as in Theorem 2, suppose that either $\lambda \ge 0$ or $0<\sigma <{\displaystyle \frac{2}{d-2}}$. Then, for any given $T_0>0$ and for any stopping time τ with $\tau <inf(T_0,{\tau }^{*}\left(u_0\right))$ almost surely, we have

(19) $\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}H\left(u\left(t\right)\right)\right)\le C\left(T_0,{\left|\varphi \right|}_{L_2^{0,1}}^2,\mathbf{E}\left(M\left(u_0\right)\right),\mathbf{E}\left(H\left(u_0\right)\right)\right),$

(20) $\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{2\alpha +2}dx\right)\le C\left(T_0,{\left|\varphi \right|}_{L_2^{0,1}}^2,\mathbf{E}\left(M\left(u_0\right)\right),\mathbf{E}\left(H\left(u_0\right)\right)\right).$

Proof. 

We first consider the case where $\lambda \ge 0$ and $0<\sigma \le {\displaystyle \frac{2}{d-2}}$. (15) implies

$\begin{array}{cc}\hfill \mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}H\left(u\left(t\right)\right)\right)\le & \mathbf{E}\left(H\left(u_0\right)\right)+\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}\mathrm{Im}{\int }_0^t{\int }_{{\mathbb{R}}^d}\Delta \overline{u}dxdW\right)\hfill \\ \hfill & +2\lambda \mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}\mathrm{Im}{\int }_0^t{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{2\sigma }\overline{u}dxdW\right)\hfill \\ \hfill & +(\alpha +2)\kappa \mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}\mathrm{Im}{\int }_0^t{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{\alpha }\overline{u}dxdW\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\sum _{i\in N}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}{\int }_0^t{\int }_{{\mathbb{R}}^d}{|\nabla \varphi e_i\left(x\right)|}^{2}dxds\right)\hfill \\ \hfill & +\lambda \sum _{i\in N}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}{\int }_0^t{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{2\sigma }{\left|\varphi e_i\left(x\right)\right|}^{2}dxds\right)\hfill \\ \hfill & +2\sigma \lambda \sum _{i\in N}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}{\int }_0^t{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{2\sigma -2}{\left(\mathrm{Im}\left(\overline{u}\varphi e_i\left(x\right)\right)\right)}^{2}dxds\right)\hfill \\ \hfill & +{\displaystyle \frac{\alpha +2}{2}}\kappa \sum _{i\in N}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}{\int }_0^t{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{\alpha }{\left|\varphi e_i\left(x\right)\right|}^{2}dxds\right)\hfill \\ \hfill & +{\displaystyle \frac{\alpha (\alpha +2)}{2}}\kappa \sum _{i\in N}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}{\int }_0^t{\int }_{{\mathbb{R}}^d}{\left|u\right|}^{\alpha -2}{\left(\mathrm{Im}\left(\overline{u}\varphi e_i\left(x\right)\right)\right)}^{2}dxds\right)\hfill \\ \hfill & :=\sum _{k=1}^9{I}_k.\hfill \end{array}$

We estimate $I_k$ one by one; using the martingale inequality, we have

$\begin{array}{cc}\hfill I_2+I_3+I_4\le 3\mathbf{E}\left({\left({\int }_0^{\tau }|{\varphi }^{*}\Delta \overline{u}{|}_{L^2}^{2}dt\right)}^{{\displaystyle \frac{1}{2}}}\right)& +6\lambda \mathbf{E}\left({\left({\int }_0^{\tau }|{\varphi }^{*}{|u|}^{2\sigma }\overline{u}{|}_{L^2}^{2}dt\right)}^{{\displaystyle \frac{1}{2}}}\right)\hfill \\ \hfill & +3(\alpha +2)\kappa \mathbf{E}\left({\left({\int }_0^{\tau }|{\varphi }^{*}{|u|}^{\alpha }\overline{u}{|}_{L^2}^{2}dt\right)}^{{\displaystyle \frac{1}{2}}}\right).\hfill \end{array}$

Note that ${\varphi }^{*}$ is bounded from $H^{-1}$ into $L^2$, then through the Young inequality we have

$I_2\le 3\mathbf{E}\left({\left({\int }_0^{\tau }|{\varphi }^{*}{|}_{L_2^{0,1}}^2|\nabla u{|}_{L^2}^{2}dt\right)}^{{\displaystyle \frac{1}{2}}}\right)\le {\displaystyle \frac{1}{16}}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}|\nabla u{|}_{L^2}^2\right)+C(T_0,|\varphi {|}_{L_2^{0,1}}^2).$

Since $H^1$ can be embedded into $L^{{\displaystyle \frac{2\sigma +2}{2\sigma +1}}}$($L^{{\displaystyle \frac{\alpha +2}{\alpha +1}}}$, respectively), similarly, we can obtain

$\begin{array}{cc}\hfill I_3\le 6\lambda \mathbf{E}\left({\left({\int }_0^{\tau }|{\varphi }^{*}{|}_{L_2^{0,1}}^2|{|u|}^{2\sigma +1}{|}_{L^{{\displaystyle \frac{2\sigma +2}{2\sigma +1}}}}^{2}dt\right)}^{{\displaystyle \frac{1}{2}}}\right)\le & {\displaystyle \frac{\lambda }{4(\sigma +1)}}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}|u{|}_{L^{2\sigma +2}}^{2\sigma +2}\right)\hfill \\ \hfill & +C(T_0,\lambda ,|\varphi {|}_{L_2^{0,1}}^2),\hfill \end{array}$

and

$\begin{array}{cc}\hfill I_4\le 3(\alpha +2)\kappa \mathbf{E}\left({\left({\int }_0^{\tau }|{\varphi }^{*}{|}_{L_2^{0,1}}^2|{|u|}^{\alpha +1}{|}_{L^{{\displaystyle \frac{\alpha +2}{\alpha +1}}}}^{2}dt\right)}^{{\displaystyle \frac{1}{2}}}\right)\le & {\displaystyle \frac{1}{16}}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}|u{|}_{L^{\alpha +2}}^{\alpha +2}\right)\hfill \\ \hfill & +C(T_0,\kappa ,|\varphi {|}_{L_2^{0,1}}^2).\hfill \end{array}$

$I_5$ is easily estimated as follows

$I_5\le {\displaystyle \frac{1}{2}}\sum _{i\in N}{\int }_0^{\tau }{\int }_{{\mathbb{R}}^d}{|\nabla \varphi e_i\left(x\right)|}^{2}dxdt\le {\displaystyle \frac{1}{2}}|\varphi {|}_{L_2^{0,1}}^2{T}_0.$

In terms of $I_6-I_9$, we have

$\begin{array}{cc}\hfill I_6+I_7\le & (2\sigma +1)\lambda \sum _{i\in N}\mathbf{E}\left({\int }_0^{\tau }{\int }_{{\mathbb{R}}^d}{|u|}^{2\sigma }{|\varphi e_i\left(x\right)|}^{2}dxdt\right)\hfill \\ \hfill \le & (2\sigma +1)\lambda \sum _{i\in N}\mathbf{E}\left({\int }_0^{\tau }|u{|}_{L^{2\sigma +2}}^{2\sigma }|\varphi e_i\left(x\right){|}_{L^{2\sigma +2}}^{2}dt\right)\hfill \\ \hfill \le & {\displaystyle \frac{\lambda }{4(\sigma +1)}}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}|u{|}_{L^{2\sigma +2}}^{2\sigma +2}\right)+C(T_0,\lambda ,|\varphi {|}_{L_2^{0,1}}^2)\hfill \end{array}$

by using the Hölder and Young inequalities. By the analogous arguments

$\begin{array}{cc}\hfill I_8+I_9& \le {\displaystyle \frac{(\alpha +2)(\alpha +1)}{2}}\kappa \sum _{i\in N}\mathbf{E}\left({\int }_0^{\tau }{\int }_{{\mathbb{R}}^d}{|u|}^{\alpha }{|\varphi e_i\left(x\right)|}^{2}dxdt\right)\hfill \\ \hfill & \le {\displaystyle \frac{1}{16}}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}|u{|}_{L^{\alpha +2}}^{\alpha +2}\right)+C(T_0,\kappa ,|\varphi {|}_{L_2^{0,1}}^2).\hfill \end{array}$

Thus, we finally achieve

$\begin{array}{cc}\hfill \mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}H\left(u\left(t\right)\right)\right)\le & \mathbf{E}\left(H\left(u_0\right)\right)+{\displaystyle \frac{1}{16}}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}|\nabla u{|}_{L^2}^2\right)+{\displaystyle \frac{\lambda }{2(\sigma +1)}}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}|u{|}_{L^{2\sigma +2}}^{2\sigma +2}\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{8}}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}|u{|}_{L^{\alpha +2}}^{\alpha +2}\right)+C(T_0,\lambda ,\kappa ,|\varphi {|}_{L_2^{0,1}}^2).\hfill \end{array}$

Together with (16), we obtain (19) as $\lambda \ge 0$.

We now consider the case where $\lambda <0$ and $0<\sigma <{\displaystyle \frac{2}{d-2}}$, we have

$\begin{array}{cc}\hfill \mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}H\left(u\left(t\right)\right)\right)\le & \mathbf{E}\left(H\left(u_0\right)\right)+\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}\mathrm{Im}{\int }_0^t{\int }_{{\mathbb{R}}^d}\Delta \overline{u}dxdW\right)\hfill \\ \hfill & +2|\lambda |\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}\mathrm{Im}{\int }_0^t{\int }_{{\mathbb{R}}^d}{|u|}^{2\sigma }\overline{u}dxdW\right)\hfill \\ \hfill & +(\alpha +2)\kappa \mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}\mathrm{Im}{\int }_0^t{\int }_{{\mathbb{R}}^d}{|u|}^{\alpha }\overline{u}dxdW\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\sum _{i\in N}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}{\int }_0^t{\int }_{{\mathbb{R}}^d}{|\nabla \varphi e_i\left(x\right)|}^{2}dxds\right)\hfill \\ \hfill & +{\displaystyle \frac{\alpha +2}{2}}\kappa \sum _{i\in N}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}{\int }_0^t{\int }_{{\mathbb{R}}^d}{|u|}^{\alpha }{|\varphi e_i\left(x\right)|}^{2}dxds\right)\hfill \\ \hfill & +{\displaystyle \frac{\alpha (\alpha +2)}{2}}\kappa \sum _{i\in N}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}{\int }_0^t{\int }_{{\mathbb{R}}^d}{|u|}^{\alpha -2}{\left(\mathrm{Im}\left(\overline{u}\varphi e_i\left(x\right)\right)\right)}^{2}dxds\right)\hfill \\ \hfill & +\mathbf{E}(\underset{t\in [0,\tau ]}{sup}(2a|\lambda |{\int }_0^t{\int }_{{\mathbb{R}}^d}{|u|}^{2\sigma +\alpha +2}dxds\hfill \\ \hfill & -a(\alpha +2)\kappa {\int }_0^t{\int }_{{\mathbb{R}}^d}{|u|}^{2\alpha +2}dxds\left)\right)\hfill \\ \hfill & =I_1+I_2+I_3+I_4+I_5+I_8+I_9+I_{10}.\hfill \end{array}$

Only the term $I_{10}$ needs to be estimated, through the interpolation and Young inequality, we have

$|u{|}_{L^{2\sigma +\alpha +2}}^{2\sigma +\alpha +2}\le |u{|}_{L^{2\alpha +2}}^{(2\sigma +\alpha +2){\theta }^{\prime }}|u{|}_{L^{\alpha +2}}^{(2\sigma +\alpha +2)(1-{\theta }^{\prime })}\le \epsilon |u{|}_{L^{2\alpha +2}}^{2\alpha +2}+C\left(\epsilon \right){|u|}_{L^{\alpha +2}}^{2\sigma +\alpha +2}$

with ${\theta }^{\prime }={\displaystyle \frac{\sigma (d+2)(d-2)}{2\left(\right(d-2)\sigma +d)}}\in (0,1)$ and sufficiently small $\epsilon$.

Then,

$I_{10}\le 2a|\lambda |C\left(\epsilon \right)\mathbf{E}\left({\int }_0^{\tau }{|u|}_{L^{\alpha +2}}^{2\sigma +\alpha +2}dt\right)\le {\displaystyle \frac{1}{8}}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}|u{|}_{L^{\alpha +2}}^{2\alpha +2}\right)+C(T_0,\lambda ).$

In summary, when $\lambda <0$, we have

$\begin{array}{cc}\hfill \mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}H\left(u\left(t\right)\right)\right)\le & \mathbf{E}\left(H\left(u_0\right)\right)+{\displaystyle \frac{1}{16}}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}|\nabla u{|}_{L^2}^2\right)+{\displaystyle \frac{|\lambda |}{4(\sigma +1)}}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}|u{|}_{L^{2\sigma +2}}^{2\sigma +2}\right)\hfill \end{array}$

$\begin{array}{cc}\hfill & +{\displaystyle \frac{1}{4}}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}|u{|}_{L^{\alpha +2}}^{\alpha +2}\right)+C(T_0,|\lambda |,\kappa ,|\varphi {|}_{L_2^{0,1}}^2)\hfill \\ \hfill \le & \mathbf{E}\left(H\left(u_0\right)\right)+{\displaystyle \frac{1}{16}}\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}|\nabla u{|}_{L^2}^2\right)+C(|\lambda |,\epsilon )\mathbf{E}\left(\underset{t\in [0,\tau ]}{sup}|u{|}_{L^{\alpha +2}}^{\alpha +2}\right)\hfill \\ \hfill & +C(T_0,|\lambda |,\kappa ,|\varphi {|}_{L_2^{0,1}}^2,|u_0{|}_{L^2}^2),\hfill \end{array}$

where substituting (18) into $\mathbf{E}\left({sup}_{t\in [0,\tau ]}|u{|}_{L^{2\sigma +2}}^{2\sigma +2}\right)$ and combining with Lemma 4. Thus, we deduce (19) as $\lambda <0$. Then, (20) follows. □

Together with Lemma 5, we have proved Theorem 1.

Author Contributions

Methodology, L.M.; formal analysis, L.M.; writing—original draft preparation, L.Q.; writing—review and editing, L.Q.; project administration, L.M.; funding acquisition, L.M. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

No new data were created or analyzed in this study.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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